1 Introduction
When data has up to three variables, visualization is relatively intuitive, while with more than three variables, we face the challenge of visualizing high dimensions on 2D displays. This issue was tackled by the grand tour (Asimov 1985) which can be used to view data in more than three dimensions using linear projections. It is based on the idea of rotations of a lower-dimensional projection in high-dimensional space. The grand tour allows users to see dynamic low-dimensional (typically 2D) projections of higher dimensional space. Originally, Asimov’s grand tour presents the viewer with an automatic movie of projections with no user control. Since then new work has added interactivity to the tour, giving more control to users (Buja et al. 2005). New variations include the manual (Cook and Buja 1997) or radial tour (Laa et al. 2023), little tour, guided tour (Cook et al. 1995), local tour, and planned tour. These are different ways of selecting the sequence of projection bases for the tour, for an overview see Lee et al. (2022).
The guided tour combines projection pursuit with the grand tour and it is implemented in the tourr package (Wickham et al. 2011). Projection pursuit is a procedure used to locate the projection of high-to-low dimensional space that should expose the most interesting feature of data, originally proposed in Kruskal (1969). It involves defining a criterion of interest, a numerical objective function that indicates the interestingness of each projection, and an optimization for selecting planes with increasing values of the function. In the literature, a number of such criteria have been developed based on clustering, spread, and outliers.
A tour path is a sequence of projections and we use an interpolation to produce small steps simulating a smooth movement. The current implementation of tour in the tourr package uses geodesic interpolation between planes. The geodesic interpolation path is the shortest path between planes with no within-plane spin (see Buja et al. (2004) for more details). As a result, the rendered target plane could be a within-plane rotation of the target plane originally specified. This is not a problem when the structure we are looking for can be identified from any rotation. However, even simple associations in 2D, such as the calculated correlation between variables, can be very different when the basis is rotated.
Most projection pursuit indexes, particularly those provided by in tourr are rotationally invariant. However, there are some where the orientation of frames does matter. One example is the splines index proposed by Grimm (2016). The splines index computes a spline model for the two variables in a projection, in order to measure non-linear association. It compares the variance of the response variable to the variance of residuals, and the functional dependence is stronger when the index value is larger. It can be useful to detect non-linear relationships in high-dimensional data. However, its value will change substantially if the projection is rotated within the plane (Laa and Cook 2020). The procedure in Grimm (2016) was less affected by the orientation because it considered only pairs of variables, and it selects the maximum value found when exchanging which variable is considered as predictor and response variable.
Figure 1 illustrates the rotational invariance problem for a modified splines index, where we always consider the horizontal direction as the predictor variable, and the vertical direction as the response. Thus, our modified index computes the splines on one orientation, exaggerating the rotational variability. The example data was simulated to follow a sine curve and the modified splines index is calculated on different within-plane rotations of the data. Although they have the same structure, the index values vary greatly.
The lack of rotation invariance of the splines index raises complications in the optimization process in the projection-pursuit guided tour as available in tourr. Fixing this is the motivation of this work. The goal with the frame-to-frame interpolation is that optimization would find the best within-plane rotation, and thus appropriately optimize the index.
Figure 1: The impact of rotation on a spline index that is NOT rotation invariant. The index value for different within-plane rotations take very different values: (a) original projection has maximum index value of 1.00, (b) axes rotated 45\(^o\) drops index value to 0.83, (c) axes rotated 60\(^o\) drops index to a very low 0.26. Geodesic interpolation between planes will have difficulty finding the maximum of an index like this because it is focused only on the projection plane, not the frame defining the plane.
A few alternatives to geodesic interpolation were proposed by Buja et al. (2005) including the decomposition of orthogonal matrices, Givens decomposition, and Householder decomposition. The purpose of the woylier package is to implement the Givens paths method in R. This algorithm adapts the Given’s matrix decomposition technique which allows the interpolation to be between frames rather than planes.
This article is structured as follows. The next section provides the theoretical framework of the Givens interpolation method followed by a section about the implementation in R. The method is applied to search for nonlinear associations between currency cross-rates.
2 Background
The tour method of visualization shows a movie that is an animated high-to-low dimensional data rotation. It is a one-parameter (time) family of static projections. Algorithms for such dynamic projections are based on the idea of smoothly interpolating a discrete sequence of projections (Buja et al. 2005).
The topic of this article is the construction of the paths of projections. The interpolation of these paths can be compared to connecting line segments that interpolate points in Euclidean space. Interpolation acts as a bridge between a continuous animation and the discrete choice of sequences of projections.
Interpolating paths of planes versus paths of frames
The tourr package implements geodesic interpolation between planes, and the final interpolation step will reach the rotation of the target frame that is avoiding any within-plane spin along the path. When the the orientation of projections matters interpolation between frames is required. The orientation of the frames could be important when a non-linear projection pursuit index function is used in the guided tour. This is illustrated by the different index values shown in the sketch in Figure 2, as well as the splines index for the sine curve in Figure 1.
XXX I like the dog sketch but currently seems not really needed - what would be nice would be showing a bit more here: starting plane is the dog looking in a different direction and also at an angle, target is the one aligned as currently seen in the front, geodesic path gets us to the target view at the angle as seen in the starting plane, givens path gives the exact target. Not sure how difficult it would be to make this though…
Figure 2: Plane to plane interpolation (left) and Frame to frame interpolation (right). We used dog index for illustration purposes. For some non-linear index orientation of data could affect the index.
To describe the interpolation algorithms we will use the following notation.
Let \(p\) be the dimension of original data and \(d\) be the dimension onto which the data is being projected.
A frame \(F\) is defined as a \(p\times d\) matrix with pairwise orthogonal columns of unit length that satisfies \[F^TF = I_d,\] where \(I_d\) is the identity matrix in d dimensions.
Paths of frames are given by continuous one-parameter families \(F(t)\) where \(t\in [a, z]\) represents time. We denote the starting frame (at time \(a\)) by \(F_a = F(a)\) and target frame (at time \(z\)) by \(F_z = F(z)\). Usually, \(F_z\) is the target frame that has been chosen according to the selected tour method. While a grand tour chooses target frames randomly, the guided tour chooses the target frame by optimizing the projection pursuit index. Interpolation methods are used to find the path that moves from \(F_a\) to \(F_z\).
Preprojection algorithm
In order to make the interpolation algorithm simple, we carry out a preprojection step to find the subspace that the interpolation path, \(F(t)\), is traversing. In other words, the preprojection step is defining the joint subspace of \(F_a\) and \(F_z\) and makes sure the interpolation path is limited to that space.
The procedure starts with forming an orthonormal basis by applying Gram-Schmidt to \(F_z\) with regards to \(F_a\), i.e. we find the \(p\times d\) matrix that contains the component of \(F_z\) that is orthogonal to \(F_a\). We denote this orthonormal basis by \(F_\star\). Then we build the preprojection basis \(B\) by combining \(F_a\) and \(F_\star\) as follows:
\[B = (F_a, F_{\star})\]
The dimension of the resulting orthonormal basis, \(B\), is \(p\times 2d\).
Then, we can express the original frames in terms of this basis:
\[F_a = B W_a, F_z = B W_z\]
The interpolation problem is then reduced to the construction of paths of frames \(W(t)\) that interpolate between the preprojected frames \(W_a\) and \(W_z\). By construction \(W_a\) is a \(2d\times d\) matrix of 1s and 0s. This is an important characteristic for our interpolation algorithm of choice, the Givens interpolation.
Givens interpolation path algorithm
A rotation matrix is a transformation matrix used to perform a rotation in Euclidean space. The matrix that rotates a 2D plane by an angle \(\theta\) looks like this:
\[ \begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix} \]
If the rotation is in the plane of two selected variables, it is called a Givens rotation. Let’s denote those 2 variables as \(i\) and \(j\). The Givens rotation is used for introducing zeros, for example when computing the QR decomposition of a matrix in linear algebra problems.
The interpolation method in the woylier package is based on the fact that in any vector of a matrix, one can zero out the \(i\)-th coordinate with a Givens rotation in the \((i, j)\)-plane for any \(j\neq i\) (Golub and Loan 1989). This rotation affects only coordinates \(i\) and \(j\) and leaves all other coordinates unchanged. Sequences of Givens rotations can map any orthonormal d-frame \(F\) in p-space to the standard d-frame \[E_d=((1, 0, 0, ...)^T, (0, 1, 0, ...)^T, ...).\]
The resulting interpolation path construction algorithm from starting frame \(F_a\) to target frame \(F_z\) is illustrated below. The example is for \(p=6\) and \(d=2\).
- Construct preprojection basis \(B\) by orthonormalizing \(F_z\) with regards tp \(F_a\) with Gram-Schmidt.
In our example, \(F_a\) and \(F_z\) are \(p\times d\) or \(6\times2\) matrices that are orthonormal. The preprojection basis \(B\) is \(p\times 2d\) matrix that is \(6\times 4\).
- Get the preprojected frames using the preprojection basis \(B\). \[W_a = B^TF_a = E_d\] and \[W_z = B^TF_z\]
In our example, \(W_a\) looks like:
\[ \begin{bmatrix}1 & 0 \\0 &1 \\ 0&0 \\0&0\end{bmatrix} \]
\(W_z\) is an orthonormal \(2d\times d\) matrix that looks like:
\[ \begin{bmatrix} a_{11} & a_{12} \\a_{21} &a_{22} \\ a_{31}&a_{32} \\a_{41}&a_{42}\end{bmatrix} \]
- Then, we can construct a sequence of Givens rotations that maps \(W_z\) to \(W_a\) with such angles that makes one element zero at a time:
\[ W_a = R_m(\theta_m) ... R_2(\theta_2)R_1(\theta_1)W_z\]
At each rotation, the angle \(\theta_i\) that zero out the next coordinate of a plane is calculated. Here \(m = \sum_{k=1}^d (2d - k)\), so when \(d=2\) we need \(m=5\) rotations with 5 different angles, each making one element 0. For example, the first rotation angle \(\theta_1\) is an angle in radians between \((1, 0)\) and \((a_{11}, a_{21})\). This rotation matrix would make element \(a_{21}\) zero:
\[R_1(\theta_1) = G(1, 2, \theta_1) = \begin{bmatrix} cos\theta_1 & -sin\theta_1 & 0 & 0 \\sin\theta_1 &cos\theta_1 & 0 &0 \\ 0&0&1&0 \\0&0&0&1\end{bmatrix}\] Here \(G(i,j,\theta_k)\) denotes a Givens rotation in components \(i\) and \(j\) by angle \(\theta_k\). In the same way we zero out the elements \(a_{31}\) and \(a_{41}\). Because of the orthonormality this means that now \(a_{11} = 1\) and that \(a_{12} = 0\). We thus need only two more rotations to zero out \(a_{32}\) and \(a_{42}\).
XXX something about how we calculate theta?
- The inverse mapping is obtained by reversing the sequence of rotations with the negative of the angles, we starts from the starting basis and end at the target basis.
\[R(\theta) = R_1(-\theta_1) ... R_m(-\theta_m), \ W_z = R(\theta)W_a\]
Performing these rotations would go from the starting frame to the target frame in one step. But we want to do it sequentially in a number of steps so interpolation between frames looks dynamic.
Next we include the time parameter, \(t\), so that the interpolation process can be rendered in the movie-like sequence. We break each \(\theta_k\) into the number of steps, \(n_{step}\), that we want to take from the starting frame to the target frame, which means it moves by equal angle in each step. Here \(n_{step}\) should vary based on the angular distance between \(F_a\) and \(F_z\), such that when watching a sequence of interpolations we have a fixed angular speed.
Finally, we reconstruct our original frames using \(B\). This reconstruction is done at each step of interpolation so that we have the interpolated path of frames as the result.
\[F_t = B W_t\]
At each time \(t\) we can project the data using the frame \(F_t\).
3 Implementation
We implemented each of the steps in the Givens interpolation path algorithm in separate functions and combined them into a single function givens_full_path() to produce the full set of \(F_t\). The same functions are used to integrate the Givens interpolation with the animate() functions of the tourr package. Table 1 lists the input and output of each functions and it’s descriptions, functions to use with animate() are described separately below.
| name | description | input | output |
|---|---|---|---|
givens_full_path(Fa, Fz, nsteps)
|
Construct full set of interpolated frames. | Starting and target frame (Fa, Fz) and number of steps | An array with nsteps matrices. Each matrix is an interpolated frame in between starting and target frames. |
preprojection(Fa, Fz)
|
Build the pre-projection space by orthonormalizing Fz with regard to Fa. | Starting and target frame (Fa, Fz) | B pre-projection p x 2d matrix |
construct_preframe(F, B)
|
Construct preprojected frames. | An orthonormal frame F and the pre-projection matrix B | Pre-projected frame in pre-projection space |
row_rot(A, i, j, theta)
|
Performs Givens rotation on the matrix A. | Matrix A, components to rotate (i, j) and rotation angle theta | Input matrix after Givens rotation |
calculate_angles(Wa, Wz)
|
Calculate angles of required rotations to map Wz to Wa. | Pre-projected frames (Wa, Wz) | Named list of angles |
construct_moving_frame(Wt, B)
|
Reconstruct interpolated frames using pre-projection. | Pre-projection matrix B, frame of givens path | A frame of a step in the interpolation |
When using tourr we typically want to run a tour live, such that target selection and interpolation are interleaved, and the display will show the data for each frame \(F_t\) in the interpolation path. The implementation in tourr was described in Wickham et al. (2011), and with woylier we provide functions to use the Givens interpolation with the grand tour, guided tour and planned tour. To do this we rely primarily on the function givens_info() which calls the functions listed in the Table above and collects all necessary information for interpolating between a given starting and target frame. The function givens_path() then defines the interpolation and can be used instead of tourr::geodesic_path(). Wrapper functions for the different tour types are available to use this interpolation, since in the tourr::grand_tour() and other path functions this is fixed to use the geodesic interpolation. Calling a grand tour with Givens interpolation for direct animation will then use:
tourr::animate_xy(<data>, tour_path = woylier::grand_tour_givens())4 Comparison of geodesic interpolation and Givens interpolation
The givens_full_path() function returns the intermediate interpolation step projections for a given number of steps. The code chunk below demonstrates the interpolation between 2 random bases in 5 steps.
givens_full_path(base1, base2, nsteps = 5)To compare the path generated with the Givens interpolation to that found with geodesic interpolations we look at the rotation of the sine data shown in Figure 1. We consider a subset in \(p=4\) dimensions where the first two dimensions contain noise and the last two contain the sine curve. Starting from a random projection we want to interpolate towards the original sine curve. The path comparison is shown in Figure 3.
Figure 3: Comparison of Givens path and geodesic path between 2D projections. The Givens path preserves the frame ending at the provided basis (frame), while geodesic is agnostic to the particular basis. In general the geodesic is preferred because it removes within-plane spin, but occasionally it is helpful to very specifically arrive at the prescribed basis.
Plotting the interpolated paths
For further comparison and to check that the interpolation is moving in equally sized steps we directly plot the interpolated paths. The space of 1D projections defines a unit sphere, while 2D projections define a torus. To illustrate the space, points on the surface of the sphere and the torus shape are randomly generated by functions from the geozoo package (Schloerke 2016). The interpolated paths are then compared within that space.
A 1D projection of data in \(p\) dimensions corresponds to a linear combination where the weights are normalized. Therefore, we can plot the point on the surface of a hypersphere. In this case the Givens interpolation will reach the exact point, while geodesic interpolation might flip the direction and reach a point on the opposite side of the hypersphere. Figure 4 (left) shows the comparison of the interpolation steps using the same target plane, for an example with \(p=3\). Because the flipped target is close to the starting plane, the geodesic path is a lot shorter.
